On eigen-structure of a nonlinear map in Rn

A nonlinear eigenvalue problem for a cubic perturbation of an irreducible Stieltjes matrix is considered. It is shown that for any fixed eigenvalue the number of eigenvectors is finite with the upper bound given by 3n. The lower bound on the number of eigenvectors depends on the position of the eigenvalue of the nonlinear equation relative to eigenvalues of the Stieltjes matrix. This study is partially motivated by the analysis of discretized Gross-Pitaevskii equations which play a role in modeling of the Bose-Einstein condensation of matter at near absolute zero temperatures. In addition to standard matrix techniques, results from Lusternik-Schnirelmann category theory, from Groebner basis theory, and from Degree Theory are used.